Skip to main content
Log in

The quark-gluon vertex in Landau gauge bound-state studies

  • Regular Article - Theoretical Physics
  • Published:
The European Physical Journal A Aims and scope Submit manuscript

Abstract

We present a practical method for the solution of the quark-gluon vertex for use in Bethe-Salpeter and Dyson-Schwinger calculations. The efficient decomposition into the necessary covariants is detailed, with the numerical algorithm outlined for both real and complex Euclidean momenta. A truncation of the quark-gluon vertex, that neglects explicit back-coupling to enable the application to bound-state calculations, is given together with results for the quark propagator and quark-gluon vertex for different quark flavours. The relative impact of the various components of the quark-gluon vertex is highlighted with the flavour dependence of the effective quark-gluon interaction obtained, thus providing insight for the construction of phenomenological models within the rainbow ladder. Finally, we solve the corresponding Green's functions for complex Euclidean momenta as required in future bound-state calculations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. H.J. Munczek, Phys. Rev. D 52, 4736 (1995) hep-th/9411239.

    Article  ADS  Google Scholar 

  2. A. Bender, C.D. Roberts, L. Von Smekal, Phys. Lett. B 380, 7 (1996) nucl-th/9602012.

    Article  ADS  Google Scholar 

  3. H.X. He, F.C. Khanna, Y. Takahashi, Phys. Lett. B 480, 222 (2000).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. M.R. Pennington, R. Williams, J. Phys. G 32, 2219 (2006) hep-ph/0511254.

    Article  ADS  Google Scholar 

  5. S.X. Qin, L. Chang, Y.X. Liu, C.D. Roberts, S.M. Schmidt, Phys. Lett. B 722, 384 (2013) arXiv:1302.3276 [nucl-th].

    Article  ADS  MATH  Google Scholar 

  6. C.D. Roberts, A.G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994) hep-ph/9403224.

    Article  ADS  Google Scholar 

  7. R. Alkofer, L. von Smekal, Phys. Rep. 353, 281 (2001) hep-ph/0007355.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. P. Maris, C.D. Roberts, Int. J. Mod. Phys. E 12, 297 (2003) nucl-th/0301049.

    Article  ADS  Google Scholar 

  9. C.S. Fischer, J. Phys. G 32, R253 (2006) hep-ph/0605173.

    Article  ADS  Google Scholar 

  10. A. Bashir, L. Chang, I.C. Cloet, B. El-Bennich, Y.X. Liu, C.D. Roberts, P.C. Tandy, Commun. Theor. Phys. 58, 79 (2012) arXiv:1201.3366 [nucl-th].

    Article  ADS  MATH  Google Scholar 

  11. C.D. Roberts, arXiv:1203.5341 [nucl-th].

  12. P. Maris, C.D. Roberts, Phys. Rev. C 56, 3369 (1997) nucl-th/9708029.

    Article  ADS  Google Scholar 

  13. P. Maris, P.C. Tandy, Phys. Rev. C 60, 055214 (1999) nucl-th/9905056.

    Article  ADS  Google Scholar 

  14. P. Maris, P.C. Tandy, Phys. Rev. C 61, 045202 (2000) nucl-th/9910033.

    Article  ADS  Google Scholar 

  15. D. Jarecke, P. Maris, P.C. Tandy, Phys. Rev. C 67, 035202 (2003) nucl-th/0208019.

    Article  ADS  Google Scholar 

  16. P. Maris, P.C. Tandy, Phys. Rev. C 65, 045211 (2002) nucl-th/0201017.

    Article  ADS  Google Scholar 

  17. P. Maris, P.C. Tandy, Nucl. Phys. Proc. Suppl. 161, 136 (2006) nucl-th/0511017.

    Article  ADS  Google Scholar 

  18. A. Krassnigg, Phys. Rev. D 80, 114010 (2009) arXiv:0909.4016 [hep-ph].

    Article  ADS  Google Scholar 

  19. A. Krassnigg, M. Blank, Phys. Rev. D 83, 096006 (2011) arXiv:1011.6650 [hep-ph].

    Article  ADS  Google Scholar 

  20. L. Chang, I.C. Cloet, C.D. Roberts, S.M. Schmidt, P.C. Tandy, Phys. Rev. Lett. 111, 141802 (2013) arXiv:1307.0026 [nucl-th].

    Article  ADS  Google Scholar 

  21. G. Eichmann, R. Alkofer, A. Krassnigg, D. Nicmorus, Phys. Rev. Lett. 104, 201601 (2010) arXiv:0912.2246 [hep-ph].

    Article  ADS  Google Scholar 

  22. G. Eichmann, Phys. Rev. D 84, 014014 (2011) arXiv:1104.4505 [hep-ph].

    Article  ADS  Google Scholar 

  23. G. Eichmann, C.S. Fischer, Eur. Phys. J. A 48, 9 (2012) arXiv:1111.2614 [hep-ph].

    Article  ADS  Google Scholar 

  24. H. Sanchis-Alepuz, G. Eichmann, S. Villalba-Chavez, R. Alkofer, Phys. Rev. D 84, 096003 (2011) arXiv:1109.0199 [hep-ph].

    Article  ADS  Google Scholar 

  25. H.L.L. Roberts, L. Chang, I.C. Cloet, C.D. Roberts, Few-Body Syst. 51, 1 (2011) arXiv:1101.4244 [nucl-th].

    Article  ADS  Google Scholar 

  26. J. Segovia, C. Chen, I.C. Cloet, C.D. Roberts, S.M. Schmidt, S. Wan, Few-Body Syst. 55, 1 (2014) arXiv:1308.5225 [nucl-th].

    Article  ADS  Google Scholar 

  27. P. Watson, W. Cassing, P.C. Tandy, Few-Body Syst. 35, 129 (2004) hep-ph/0406340.

    ADS  Google Scholar 

  28. M.S. Bhagwat, A. Holl, A. Krassnigg, C.D. Roberts, P.C. Tandy, Phys. Rev. C 70, 035205 (2004) nucl-th/0403012.

    Article  ADS  Google Scholar 

  29. C.S. Fischer, P. Watson, W. Cassing, Phys. Rev. D 72, 094025 (2005) hep-ph/0509213.

    Article  ADS  Google Scholar 

  30. H.H. Matevosyan, A.W. Thomas, P.C. Tandy, Phys. Rev. C 75, 045201 (2007) nucl-th/0605057.

    Article  ADS  Google Scholar 

  31. C.S. Fischer, R. Williams, Phys. Rev. D 78, 074006 (2008) arXiv:0808.3372 [hep-ph].

    Article  ADS  Google Scholar 

  32. R. Alkofer, C.S. Fischer, R. Williams, Eur. Phys. J. A 38, 53 (2008) arXiv:0804.3478 [hep-ph].

    Article  ADS  Google Scholar 

  33. C.S. Fischer, R. Williams, Phys. Rev. Lett. 103, 122001 (2009) arXiv:0905.2291 [hep-ph].

    Article  ADS  Google Scholar 

  34. L. Chang, C.D. Roberts, Phys. Rev. Lett. 103, 081601 (2009) arXiv:0903.5461 [nucl-th].

    Article  ADS  MathSciNet  Google Scholar 

  35. W. Heupel, T. Goecke, C.S. Fischer, Eur. Phys. J. A 50, 85 (2014) arXiv:1402.5042 [hep-ph].

    Article  ADS  Google Scholar 

  36. M. Vujinovic, R. Williams, arXiv:1411.7619 [hep-ph].

  37. J.M. Cornwall, Phys. Rev. D 26, 1453 (1982).

    Article  ADS  Google Scholar 

  38. P. Watson, R. Alkofer, Phys. Rev. Lett. 86, 5239 (2001) hep-ph/0102332.

    Article  ADS  Google Scholar 

  39. C. Lerche, L. von Smekal, Phys. Rev. D 65, 125006 (2002) hep-ph/0202194.

    Article  ADS  Google Scholar 

  40. D. Zwanziger, Phys. Rev. D 67, 105001 (2003) hep-th/0206053.

    Article  ADS  Google Scholar 

  41. J.M. Pawlowski, D.F. Litim, S. Nedelko, L. von Smekal, Phys. Rev. Lett. 93, 152002 (2004) hep-th/0312324.

    Article  ADS  Google Scholar 

  42. R. Alkofer, C.S. Fischer, F.J. Llanes-Estrada, Phys. Lett. B 611, 279 (2005) 670.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  43. C.S. Fischer, J.M. Pawlowski, Phys. Rev. D 75, 025012 (2007) hep-th/0609009.

    Article  ADS  MathSciNet  Google Scholar 

  44. A.C. Aguilar, D. Binosi, J. Papavassiliou, Phys. Rev. D 78, 025010 (2008) arXiv:0802.1870 [hep-ph].

    Article  ADS  Google Scholar 

  45. C.S. Fischer, J.M. Pawlowski, Phys. Rev. D 80, 025023 (2009) arXiv:0903.2193 [hep-th].

    Article  ADS  MathSciNet  Google Scholar 

  46. R. Alkofer, C.S. Fischer, F.J. Llanes-Estrada, Mod. Phys. Lett. A 23, 1105 (2008) hep-ph/0607293.

    Article  ADS  MATH  Google Scholar 

  47. R. Alkofer, C.S. Fischer, F.J. Llanes-Estrada, K. Schwenzer, Ann. Phys. 324, 106 (2009) arXiv:0804.3042 [hep-ph].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  48. P. Watson, W. Cassing, Few-Body Syst. 35, 99 (2004) hep-ph/0405287.

    ADS  Google Scholar 

  49. A.W. Thomas, Prog. Part. Nucl. Phys. 61, 219 (2008) arXiv:0805.4437 [hep-ph].

    Article  ADS  Google Scholar 

  50. C.S. Fischer, D. Nickel, J. Wambach, Phys. Rev. D 76, 094009 (2007) arXiv:0705.4407 [hep-ph].

    Article  ADS  Google Scholar 

  51. W. Heupel, S. Kubrak, G. Eichmann, C.S. Fischer, PoS Bormio 2013, 065 (2013) arXiv:1305.0650 [hep-ph].

    Google Scholar 

  52. P. Maris, AIP Conf. Proc. 892, 65 (2007) nucl-th/0611057.

    Article  ADS  Google Scholar 

  53. C. Popovici, P. Watson, H. Reinhardt, Phys. Rev. D 81, 105011 (2010) arXiv:1003.3863 [hep-th].

    Article  ADS  Google Scholar 

  54. C. Popovici, P. Watson, H. Reinhardt, Phys. Rev. D 83, 125018 (2011) arXiv:1103.4786 [hep-ph].

    Article  ADS  Google Scholar 

  55. M. Blank, A. Krassnigg, Phys. Rev. D 84, 096014 (2011) arXiv:1109.6509 [hep-ph].

    Article  ADS  Google Scholar 

  56. J. Skullerud, A. Kizilersu, JHEP 09, 013 (2002) hep-ph/0205318.

    Article  ADS  Google Scholar 

  57. J.I. Skullerud, P.O. Bowman, A. Kizilersu, D.B. Leinweber, A.G. Williams, JHEP 04, 047 (2003) hep-ph/0303176.

    Article  ADS  Google Scholar 

  58. A. Kizilersu, D.B. Leinweber, J.I. Skullerud, A.G. Williams, Eur. Phys. J. C 50, 871 (2007) hep-lat/0610078.

    Article  ADS  Google Scholar 

  59. A.C. Aguilar, D. Binosi, D. Ibanez, J. Papavassiliou, Phys. Rev. D 90, 6, 065027 (2014) arXiv:1405.3506 [hep-ph].

    Article  Google Scholar 

  60. M. Hopfer, A. Windisch, R. Alkofer, PoS ConfinementX, 073 (2012) arXiv:1301.3672 [hep-ph].

  61. A. Windisch, M. Hopfer, R. Alkofer, Acta Phys. Pol. Suppl. 6, 347 (2013) arXiv:1210.8428 [hep-ph].

    Article  Google Scholar 

  62. E. Rojas, J.P.B.C. de Melo, B. El-Bennich, O. Oliveira, T. Frederico, JHEP 10, 193 (2013) arXiv:1306.3022 [hep-ph].

    Article  ADS  Google Scholar 

  63. R. Williams, EPJ Web of Conferences 3, 03005 (2010) arXiv:0912.3494 [hep-ph].

    Article  Google Scholar 

  64. R. Alkofer, M.Q. Huber, K. Schwenzer, Phys. Rev. D 81, 105010 (2010) arXiv:0801.2762 [hep-th].

    Article  ADS  Google Scholar 

  65. J. Berges, Phys. Rev. D 70, 105010 (2004) hep-ph/0401172.

    Article  ADS  MathSciNet  Google Scholar 

  66. G. Eichmann, C.S. Fischer, Phys. Rev. D 87, 036006 (2013) arXiv:1212.1761 [hep-ph].

    Article  ADS  Google Scholar 

  67. D. Nicmorus, G. Eichmann, R. Alkofer, Phys. Rev. D 82, 114017 (2010) arXiv:1008.3184 [hep-ph].

    Article  ADS  Google Scholar 

  68. H. Sanchis-Alepuz, R. Alkofer, G. Eichmann, R. Williams, PoS QCD-TNT-II, 041 (2011) arXiv:1112.3214 [hep-ph].

  69. H. Sanchis-Alepuz, R. Williams, R. Alkofer, Phys. Rev. D 87, 096015 (2013) arXiv:1302.6048 [hep-ph].

    Article  ADS  Google Scholar 

  70. A. Cucchieri, A. Maas, T. Mendes, Phys. Rev. D 77, 094510 (2008) arXiv:0803.1798 [hep-lat].

    Article  ADS  Google Scholar 

  71. D. Dudal, O. Oliveira, J. Rodriguez-Quintero, Phys. Rev. D 86, 105005 (2012) arXiv:1207.5118 [hep-ph].

    Article  ADS  Google Scholar 

  72. M.Q. Huber, L. von Smekal, JHEP 04, 149 (2013) arXiv:1211.6092 [hep-th].

    Article  ADS  Google Scholar 

  73. D. Binosi, D. Ibanez, J. Papavassiliou, Phys. Rev. D 87, 125026 (2013) arXiv:1304.2594 [hep-ph].

    Article  ADS  Google Scholar 

  74. A.C. Aguilar, D. Binosi, D. Ibanez, J. Papavassiliou, Phys. Rev. D 89, 085008 (2014) arXiv:1312.1212 [hep-ph].

    Article  ADS  Google Scholar 

  75. A. Blum, M.Q. Huber, M. Mitter, L. von Smekal, Phys. Rev. D 89, 061703 (2014) arXiv:1401.0713 [hep-ph].

    Article  ADS  Google Scholar 

  76. G. Eichmann, R. Williams, R. Alkofer, M. Vujinovic, Phys. Rev. D 89, 105014 (2014) arXiv:1402.1365 [hep-ph].

    Article  ADS  Google Scholar 

  77. A.I. Davydychev, P. Osland, L. Saks, Phys. Rev. D 63, 014022 (2001) hep-ph/0008171.

    Article  ADS  Google Scholar 

  78. L. Chang, Y.X. Liu, C.D. Roberts, Phys. Rev. Lett. 106, 072001 (2011) arXiv:1009.3458 [nucl-th].

    Article  ADS  Google Scholar 

  79. A. Bender, W. Detmold, C.D. Roberts, A.W. Thomas, Phys. Rev. C 65, 065203 (2002) nucl-th/0202082.

    Article  ADS  Google Scholar 

  80. P. Maris, C.D. Roberts, P.C. Tandy, Phys. Lett. B 420, 267 (1998) nucl-th/9707003.

    Article  ADS  Google Scholar 

  81. R. Alkofer, P. Watson, H. Weigel, Phys. Rev. D 65, 094026 (2002) hep-ph/0202053.

    Article  ADS  Google Scholar 

  82. S. Strauss, C.S. Fischer, C. Kellermann, Phys. Rev. Lett. 109, 252001 (2012) arXiv:1208.6239 [hep-ph].

    Article  ADS  Google Scholar 

  83. M.Q. Huber, A. Maas, L. von Smekal, JHEP 11, 035 (2012) arXiv:1207.0222 [hep-th].

    Article  ADS  Google Scholar 

  84. C.S. Fischer, D. Nickel, R. Williams, Eur. Phys. J. C 60, 47 (2009) arXiv:0807.3486 [hep-ph].

    Article  ADS  Google Scholar 

  85. M. Gimeno-Segovia, F.J. Llanes-Estrada, Eur. Phys. J. C 56, 557 (2008) arXiv:0805.4145 [hep-th].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  86. A. Krassnigg, PoS Confinement 8, 075 (2008) arXiv:0812.3073 [nucl-th].

    Google Scholar 

  87. A. Windisch, M.Q. Huber, R. Alkofer, Acta Phys. Pol. Suppl. 6, 887 (2013) arXiv:1304.3642 [hep-ph].

    Article  Google Scholar 

  88. L. von Smekal, K. Maltman, A. Sternbeck, Phys. Lett. B 681, 336 (2009) arXiv:0903.1696 [hep-ph].

    Article  ADS  Google Scholar 

  89. C.S. Fischer, A. Maas, J.M. Pawlowski, Ann. Phys. 324, 2408 (2009) arXiv:0810.1987 [hep-ph].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  90. S.K. Kim, M. Baker, Nucl. Phys. B 164, 152 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  91. J.S. Ball, T.W. Chiu, Phys. Rev. D 22, 2550 (1980) 23.

    Article  ADS  Google Scholar 

  92. H. Sanchis-Alepuz, R. Williams, arXiv:1503.05896 [hep-ph].

  93. G. Eichmann, Acta Phys. Pol. Suppl. 7, 597 (2014) arXiv:1404.4149 [nucl-th].

    Article  Google Scholar 

  94. P.O. Bowman, U.M. Heller, D.B. Leinweber, M.B. Parappilly, A.G. Williams, J.b. Zhang, Phys. Rev. D 71, 054507 (2005) hep-lat/0501019.

    Article  ADS  Google Scholar 

  95. L. Chang, C.D. Roberts, Phys. Rev. C 85, 052201 (2012) arXiv:1104.4821 [nucl-th].

    Article  ADS  Google Scholar 

  96. I.C. Cloet, C.D. Roberts, Prog. Part. Nucl. Phys. 77, 1 (2014) arXiv:1310.2651 [nucl-th].

    Article  ADS  Google Scholar 

  97. C.S. Fischer, S. Kubrak, R. Williams, Eur. Phys. J. A 50, 126 (2014) arXiv:1406.4370 [hep-ph].

    Article  ADS  Google Scholar 

  98. H. Sanchis-Alepuz, R. Williams, arXiv:1504.0776 [hep-ph].

  99. M. Mitter, J.M. Pawlowski, N. Strodthoff, Phys. Rev. D 91, 054035 (2015) arXiv:1411.7978 [hep-ph].

    Article  ADS  Google Scholar 

  100. J.S. Ball, T.W. Chiu, Phys. Rev. D 22, 2542 (1980).

    Article  ADS  Google Scholar 

  101. A. Kizilersu, M. Reenders, M.R. Pennington, Phys. Rev. D 52, 1242 (1995) hep-ph/9503238.

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Richard Williams.

Additional information

Communicated by H. Wittig

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Williams, R. The quark-gluon vertex in Landau gauge bound-state studies. Eur. Phys. J. A 51, 57 (2015). https://doi.org/10.1140/epja/i2015-15057-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epja/i2015-15057-4

Keywords

Navigation